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%I #11 Feb 19 2021 20:10:00
%S 0,0,12,216,3500,58494,1028167,18954072,363991752,7231521650,
%T 147777013109,3091874792274,65993049570175,1432803420182428,
%U 31570847522072400,704668366087255200,15907964778448807820
%N Number of orbits of length n under the map whose periodic points are counted by A061694.
%C Old name was: A061694 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n under that map.
%H Vaclav Kotesovec, <a href="/A091266/b091266.txt">Table of n, a(n) for n = 1..200</a>
%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/SIXDENIERS/bell.html">Extended Bell and Stirling Numbers From Hypergeometric Exponentiation</a>, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
%H Thomas Ward, <a href="http://web.archive.org/web/20081002082625/http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>. <a href="/A091112/a091112.pdf">[local copy]</a>.
%F If b(n) is the n-th term of A061694, then a(n) = (1/n)*Sum_{d|n}mu(d)b(n/d).
%F a(n) ~ 3^(3*n + 1) / (8 * Pi^2 * n^3). - _Vaclav Kotesovec_, Sep 05 2019
%e b(1)=0, b(3)=36 so a(3)=12.
%t Table[Sum[MoebiusMu[d] * Sum[Sum[((n/d)!/(i!*j!*(n/d - i - j)!))^3/6, {i, 1, n/d - j - 1}], {j, 1, n/d}], {d, Divisors[n]}]/n, {n, 1, 20}] (* _Vaclav Kotesovec_, Sep 05 2019 *)
%Y Cf. A061694.
%K nonn
%O 1,3
%A _Thomas Ward_, Feb 24 2004
%E Name clarified by _Michel Marcus_, May 14 2015
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